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The following formula for $\pi$ was discovered by Ramanujan: $$\frac1{\pi} = \frac{2\sqrt{2}}{9801} \sum_{k=0}^\infty \frac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}}\!$$ Does anyone know how it works, or what the motivation for it is?
One of the rules of logarithms that I've been taught is: log<sub>b</sub>x is equivalent to ( log<sub>k</sub>x / log<sub>k</sub>b ) Recently I've also seen another rule that says: log<sub>b</sub>x is equivalent to ( log<sub>b</sub>k * log<sub>k</sub>x ) Are these equivalent (perhaps via some refactoring) ? ...
One of the rules of logarithms that I've been taught is: $\log_b(x)$ is equivalent to $\frac{\log_k(x)}{\log_k(b)}$. Recently I've also seen another rule that says: $\log_b(x)$ is equivalent to $\log_b(k)\log_k(x)$. Are these equivalent (perhaps via some refactoring) ?
Can anyone suggest at least verbally how it is that a torus is constructed from two cells? (Total beginner to topology, been away from math as a whole for awhile...) Thanks for any insights!
First: They are not "equivalent", they are equal. That said: remember the meaning of "$\log_b(a) = r$. It means that $r$ is the exponent to which you must raise $b$ in order to get $a$; that is, $\log_b(a)=r$ is equivalent to $b^r=a$. So... why is $\log_b(x)=\frac{\log_k(x)}{\log_k(b)}$? Because if you let $r=\...
First: They are not "equivalent", they are equal. That said: remember the meaning of "$\log_b(a) = r$. It means that $r$ is the exponent to which you must raise $b$ in order to get $a$; that is, $\log_b(a)=r$ is equivalent to $b^r=a$. So... why is $\log_b(x)=\frac{\log_k(x)}{\log_k(b)}$? Because if you let $r=\...
Could you please help me in this problem? I have 3 independent variables, $(T,H,t)$, as inputs and one output $P$ ( I have all data for these inputs and the output, done experimentally measured every hour during one year). I want to find a formula of this form: $$P = f ( T , H , t )$$ where $t$ is the time ...
Define $(a,b) < (a',b')$ if $\max(a,b) < \max(a',b')$ or $\max(a,b) = \max(a',b')$ and $b < b'$ or $\max(a,b) = \max(a',b')$ and $b = b'$ and $a < a'$. Now I want to prove that the order-type of $(\{(b,c) : \max(b,c) = a\}, <)$ is equal to $a + a + 1$, does someone have a hint how to do this? I can't find the biject...
[Euler's rotation theorem](http://en.wikipedia.org/wiki/Euler%27s_rotation_theorem) (very informally stated) says that an rotation of a sphere in three dimensional space always has two fixed points. I believe this claim generalizes to (_n_-1)-hyperspheres in _n_-dimensional space for odd _n_, but I am unable to patch t...
Ruling out a lone eigenvalue of -1 to complete this proof?
[Euler's rotation theorem](http://en.wikipedia.org/wiki/Euler%27s_rotation_theorem) (very informally stated) says that an rotation of a sphere in three dimensional space always has two fixed points. I believe this claim generalizes to (_n_-1)-hyperspheres in _n_-dimensional space for odd _n_, but I am unable to pat...
Given a bijection $f:\mathbb Z \to \mathbb Z$ where $\mathbb Z$ is the set of all integers, does there always exist two bijections $g:\mathbb Z \to \mathbb Z$ and $h:\mathbb Z \to \mathbb Z$ which satisfy $f(n) = g(n) + h(n), ~ \forall n \in \mathbb Z$ ? Evidently it does while $\mathbb Z$ is replaced with the set o...
It would be great if someone could explain me the following. If I understand the [lecture](http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/video-lectures/lecture-1-the-geometry-of-linear-equations/) correctly, given a system of, say, following equations: $x+2y-2z=1$ <br> $2x+4y+z=3$ <br> $4...
I am looking at the first two examples in Paul Garretts[ notes][1] on p-adic number theory. The first example is computing the Newton polygon of $x^5+2x^2+5$ over $\mathbb{Q}_2$. I think this is the lower convex hull of $(0,1),(1,0),(2,0.5),(3,0),(4,0),(5,1)$. So I thought the Newton polygon should just be a horizon...
Given a bijection $f:\mathbb Z \to \mathbb Z$ where $\mathbb Z$ is the set of all integers, does there always exist two bijections $g:\mathbb Z \to \mathbb Z$ and $h:\mathbb Z \to \mathbb Z$ which satisfy $f(n) = g(n) + h(n), ~ \forall n \in \mathbb Z$ ? Evidently it does while $\mathbb Z$ is replaced with the set o...
I have a list of several entities, all of each have a numerical relationship to each other that defines an abstract distance. Is there a mathematical way to plot all of these on a 2D space, turning the abstract distance into a 2D euclidian distance, and preserving all of their distances?
Given a metric function between a set of abstract points, what is the best way to plot them on a 2D space?
Can it be done? For an arbitrary quadratic field $Q[\sqrt{d}]$, it's easy to show the equations are simply $ f_x = -\sqrt{d} f_y $, where $ f : Q[\sqrt{d}] \to Q[\sqrt{d}]$. I'm working on the case of $Q[\theta]$, when $\theta$ is a root of $\theta^3 - a\theta - b$, but I'm not sure if it's even possible. Has there ...
I need a function (a curve -- preferably a simple one) that, given $n$ points of a 2D space ($R^2$) passes (interpolates) through all points in a smooth/continuous way. Found out that what I need is a [spline](http://en.wikipedia.org/wiki/Spline_curve), however cannot find one that behaves how I need. $n$-degree Bez...
Is this a probability distribution?
I randomly pick a natural number n. Assuming that I would have picked each number with the same probability, what was the probability for me to pick n before I did it?
Firstly, consider the case when we are not obliged to have the equation 0=0+0. The question becomes: given $n \geq 1$ does there exist two permutations $g,h$ of $A:=\{-n,-n+1,\ldots,n\}$ such that $n=g(i)+h(i)$ where addition is addition over the integers (and therefore is not closed on $A$). We can interpret an ...
In high school, when we talked about "combinatorics," we solely meant "mathematics of choice." For instance: - There are 10 people who want to sit around a table. In how many ways is this possible? - We have 50 balls and 20 boxes. In how many ways can we distribute balls into the boxes? - There are 10 apples and 5...
What's Combinatorial Proof/Object/etc.?
I am reading a text on maths applied to naturally occurring networks, eg. social networks. The section I am on is "Walks" networks. The text says: > Walks: which allow both nodes and > edges to be used more than once. For > example, in Figure 1, > Mary-Bob-Sue-Ramona-Bob-Mary-Joe is a > walk, but not a tra...
I know the following: - my own position - my own facing (the angle im turned) - my targets position What i would like some help with is how i calculate the shortes way to turn and the angle to turn. If i my position was (4,4) and i faced 90 degrees and my target was to face (1,1) where my target is, i know by d...
Let $G$ and $H$ be two possibly directed, non necessarily simple, vertex-labelled graphs with respective adjacency matrices $A_G$ and $A_H$ and $V(G)=V(H)$. 1) What is the name of the graph $M$ with adjacency matrix $A_M=A_HA_G$? 2) In the unlikely event that I am the first to think about using that operation, wh...
I'm a lazy type of person. I love experimenting with things before learning the theory. I like to practice, and learn from my mistakes. In 3-4 months I have a basic University exam on Calculus, and I'd like to learn it in a fun way. I have found these videos to be quite entertaining: http://www.calculus-help.com/...
Can I learn Calculus on the web, for free, using simple tutorials?
Can anybody give me a hint how to approach the following problem, please? I actually am having a hard time stating the problem. I think an example would help you understand what the problem is. Suppose I have 5 numbers each can be max 3 digits. I sum them together. What's the probability of the sum having the las...
First: They are not "equivalent", they are equal. That said: remember the meaning of "$\log_b(a) = r$". It means that $r$ is the exponent to which you must raise $b$ in order to get $a$; that is, $\log_b(a)=r$ is equivalent to $b^r=a$. So... why is $\log_b(x)=\frac{\log_k(x)}{\log_k(b)}$? Because if you let $r=...
Here is what I need to prove: >Let $d_1,d_2,...,d_n$ be a sequence of natural numbers (>0). Show that $d_i$ is a degree sequence of some tree if and only if $\sum d_i = 2(n-1)$. I know that: 1. for any graph $\sum_{v \in V}\ deg(v) = 2e$; 2. for any tree $e=v-1$. From 1 and 2 it follows that for any tree ...
What are or where can I find style guidelines for writing math?
From your fact (1) and the hypothesis $\sum d_i = 2(n-1)$, you know $e = n - 1$. You need now only prove that any connected graph with $n - 1$ edges is a tree. EDIT - Since you are going down an alternate path, I have a new suggestion. Order the $d_i$ so that $d_1 \leq d_2 \leq \ldots \leq d_n$. Construct a...
This post is related to a previous SE post [ If a 1 meter rope …. ][1] concerning average length of a smallest segment. A rope of 1m is divided into three pieces by two random points. Find the average length of the largest segment. My answer is 11/18. Here is how I do it: Here we have two independent random v...
You seem to want some explanation that does not involve vectors, subspaces spanned by columns, etc. I have to say that, in my personal opinion, vectors, subspaces, spans, etc. are the best way to think about it, but let me give it a try. Suppose you have a $3\times 3$ linear system in $x$, $y$, and $c$, $$\begin{ar...
Here is what I need to prove: >Let $d_1,d_2,...,d_n$ be a sequence of natural numbers (>0). Show that $d_i$ is a degree sequence of some tree if and only if $\sum d_i = 2(n-1)$. I know that: 1. for any graph $\sum_{v \in V}\ deg(v) = 2e$; 2. for any tree $e=v-1$. From 1 and 2 it follows that for any tree ...
I am working through a review problem asking to find the inverse of $4\ \mathrm{mod}\ {9}$ Through examples I know that I first need to verify that the gcd is equal to 1 and write it as a linear combination of 4 and 9 to find the inverse. I can do this in just one step: gcd(4,9) 9 = 2 * 4 + 1 1 = 9...
I am working through a review problem asking to find the inverse of $4\ \mathrm{mod}\ {9}$. Through examples I know that I first need to verify that the gcd is equal to 1 and write it as a linear combination of 4 and 9 to find the inverse. I can do this in just one step: gcd(4,9) 9 = 2 * 4 + 1 1 = ...
It would be great if someone could explain me the following. If I understand the [lecture](http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/video-lectures/lecture-1-the-geometry-of-linear-equations/) correctly, given a system of, say, following equations: $x+2y-2z=1$ <br> $2x+4y+z=3$ <br> $4...
You seem to want some explanation that does not involve vectors, subspaces spanned by columns, etc. I have to say that, in my personal opinion, vectors, subspaces, spans, etc. are the best way to think about it, but let me give it a try. Suppose you have a $3\times 3$ linear system in $x$, $y$, and $c$, $$\begin{ar...
Is there a way to (efficiently) enumerate all simple graphs with a given number of vertices up to isomorphism? That is, I don't care about the degeneracy of a graph, only the set of isomorphicly isomorphically unique ones of a given size.
I am working through a review problem asking to find the inverse of $4\bmod 9 $. Through examples I know that I first need to verify that the gcd is equal to 1 and write it as a linear combination of 4 and 9 to find the inverse. I can do this in just one step: gcd(4,9) 9 = 2 * 4 + 1 1 = 9 - 2 * 4...
Motivation I have a prior on a random variable $X\sim \beta(\alpha,\beta)$ but I need to transform the variable to $Y=\frac{X}{1-X}$, for use in an analysis and I would like to know the distribution of $Y$. [Wikipedia][1] states: > if $X\sim\beta(\alpha,\beta)$ then $\frac{X}{1-X} \sim\beta^\prime(\alpha,\...
What is the rate of growth of the partial sums of the reciprocals of the odd numbers?
How does a technical and theoretical study of a project of implementing an algorithm on a specific hardware architecture? Function example: %Expresion 1: y1 = exp (- (const1 + x) ^ 2 / ((conts2 ^ 2)), y2 = y1 * const3 Where x is the input variable, y2 is the output and cost1, const3 const2 a...
Suppose $g:\mathbb{R} \rightarrow [0,\infty)$ is a strictly increasing function such that $\lim g(x) = \infty$ as $x \rightarrow \infty$. Suppose $h:\mathbb{R} \rightarrow [0,\infty)$ is a strictly decreasing function such that $\lim h(x) = 0$ as $x \rightarrow \infty$. Consider the product function $f(x)=g(x)h(x)$. ...
I will be using relativistic terminology for pseudo-Riemannian manifolds as in the book by Barret O' Neil. - If one can show that the chronal future and the chronal past of some set are disjoint then why is this enough for showing that the set is achronal? I mean one can have a set such that there is precisel...
How to prove the error estimate of the Newton-iteration?
I will be using relativistic terminology for pseudo-Riemannian manifolds as in the book by Barret O' Neil. - If one can show that the chronal future and the chronal past of some set are disjoint then why is this enough for showing that the set is achronal? I mean one can have a set such that there is precisel...
What are the advantages / disadvantages of using splines, smoothed splines, and gaussian process emulators?
I found a contradiction I couldn't resolve by my self. It's about a "Uniform White Noise". Let ${x}_{t}$ be a "White Noise" i.i.d. Random Process: $ \forall t \in \mathbb{R}, \ {x}_{t} \sim U[-1, \ 1] $ If we chose to go by the PSD definition of "White Noise" (Constant all over the Frequencies) we get...
I'm confused about the concept of pivoting a Matrix element. This operation seems to be a fundamental operation which is used as part of more complex operations such as inverting a matrix. However, the definition of pivoting, and what it entails, seems to vary depending on the context. In the context of invert...
Let's say, we have an array/matrix $n \times m$ and we need to find, how many ways can we fill this array with numbers from $\{ 1, \ldots , m\cdot n \}$, but: 1) every number can be used only 1 time 2) every column and every row should be sorted in increasing order For example, if we have $n = m = 3$, there ar...
Interesting problem - array of numbers, how many solutions/ways?
For the first question, I fear that you may have not understood the definition completely. Let $p,q\in U$ be the two points. Let $\gamma$ be the future time-like curve joining $p = \gamma(0)$ and $q= \gamma(1)$. I claim that for any $s\in (0,1)$, $\gamma(s)$ is in both the future and past of $U$. Indeed, $\gamma(s)$ is...
I was rather confused about this proof I came across in Rudin Chapter 7. The premise is: If K is a compact metric space, if $f_n \in C(K)$ for $n = 1, 2, 3, ...,$ ($C(K)$ being a set of complex-valued, continuous and bounded functions), and if $\{f_n\}$ converges uniformly on $K$, then $\{f_n\}$ is equicontinuous on ...
Don't Understand Proof in Rudin Principles of Mathematical Analysis Book Thm 7.24?
The question asks me to determine if 4Z and 5Z (with standard addition) are isomorphic and if so to give the isomorphism. My attempts: What I am having difficulty with is showing a mapping that preserves the operation. IE phi(a+b) = phi(a) + phi(b). What I have so far is: phi(a+b) = phi(5a/4) + phi(5b/4)....
The answer to (B) is actually given in both Yuval Filmus' and my answers to the question about the [average length of the shortest segment][1]. It's $$\frac{1}{n} H_n,$$ where $H_n = \sum_{k=1}^n \frac{1}{k},$ i.e., the $n$th [harmonic number][2]. <HR> "Clever" is of course subjective, but here's an argument for...
Update: The current version of this answer is more intuitive (IMHO) than the previous one. See also [this similar answer][1] to a similar question. The generalization of this problem to $n$ pieces is discussed extensively in David and Nagaraja's [Order Statistics][2] (pp. 133-135, and p. 153). If $X_1, X_2, ...
Update: The current version of this answer is more intuitive (IMHO) than the previous one. See also [this similar answer][1] to a similar question. The generalization of this problem to $n$ pieces is discussed extensively in David and Nagaraja's [Order Statistics][2] (pp. 133-135, and p. 153). If $X_1, X_2, ...
The answer to (B) is actually given in both Yuval Filmus' and my answers to the question about the [average length of the shortest segment][1]. It's $$\frac{1}{n} H_n,$$ where $H_n = \sum_{k=1}^n \frac{1}{k},$ i.e., the $n$th [harmonic number][2]. <HR> "Clever" is of course subjective, but here's an argument for...
I was reading over this in my notes, but it doesnt make any sense to me: > However, suppose we ﬁx an alphabet for > writing our programs (e.g. 8- bit > ASCII). Since each individual program > is ﬁnite in length, we can put all > possible programs into a (very long) > ordered list. For any ﬁxed character > leng...
Array of numbers, how many solutions/ways?
> Justin needs to pick 17 toy animals to > give to children at a party. The > animals come in 5 kinds: dogs, > dinosaurs, cows, lizards, and ﬁsh. How > many diﬀerent ways can he choose his > set of toys Ok the answer for this is 21 choose 4. However, I dont understand how they arrived at this answer. If there a...
I was rather confused about this proof I came across in Rudin Chapter 7. The premise is: If K is a compact metric space, if $f_n \in C(K)$ for $n = 1, 2, 3, ...,$ ($C(K)$ being a set of complex-valued, continuous and bounded functions), and if $\{f_n\}$ converges uniformly on $K$, then $\{f_n\}$ is equicontinuous on ...
I tried to prove this, but was unsuccessful for a long time.. Any ideas? Prove that $(e^x)^y=e^{xy}$ using the identities, $$e^x=\sum_{n\geq0} \frac{x^n}{n!}, (1+x)^\alpha=\sum_{n\geq0} {\alpha \choose n} x^n$$
Here is one analogy which may prove helpful: there are rational numbers, which can be represented as the ratio of two integers: 1/2, 3, 5/6 etc. This can be seen as all the numbers which can be printed out by "programs" which only have access to the "division" operator. Clearly some numbers cannot be printed out by a p...
The question asks me to determine if $4\mathbb{Z}$ and $5\mathbb{Z}$ (with standard addition) are isomorphic and if so to give the isomorphism. My attempts: What I am having difficulty with is showing a mapping that preserves the operation. i.e., $\phi(a+b) = \phi(a) + \phi(b)$. What I have so far is: $\p...
I tried to prove this, but was unsuccessful for a long time.. Any ideas? Prove that $(exp(x))^y=exp(xy)$ using the identities, $$e^x=\sum_{n\geq0} \frac{x^n}{n!}, (1+x)^\alpha=\sum_{n\geq0} {\alpha \choose n} x^n$$
I tried to prove this, but was unsuccessful for a long time.. Any ideas? Prove that $(\exp(x))^y=\exp(xy)$ using the identities, $$\exp(x)=\sum_{n\geq0} \frac{x^n}{n!}, \quad (1+x)^\alpha=\sum_{n\geq0} {\alpha \choose n} x^n$$
I was reading over this in my notes, but it doesnt make any sense to me: > However, suppose we ﬁx an alphabet for > writing our programs (e.g. 8- bit > ASCII). Since each individual program > is ﬁnite in length, we can put all > possible programs into a (very long) > ordered list. For any ﬁxed character > leng...
A two dimensional space (eg. $\mathbb{R}^2$ ) could be a flat or it could be the surface of sphere (or of any shape) in a 3-dimensional space. How do we distinguish between such spaces without invoking higher dimensions.(here I am using the term 'space' in a general term and not strictly in a manner used in mathematics...
Let $X$ be a finite dimensional normed space and $X_{1}$ be a subspace of $X$. Can one always extend a linear operator $l:X_{1}\rightarrow X$ to $L:X\rightarrow X$ such that $L$ has the same operator norm as $l$? Thanks!
Can a linear operator from a normed space to itself be extended with operator norm preserved?
There are Artin $L$-series and Dirichlet $L$-series, and zeta functions for varieties and for number fields; there are a slew of objects named after Hecke... There are also various kinds of characters in these areas. I've always had a hard time keeping all of them straight. I would be very grateful if someone could ...
Let $X$ be a finite dimensional real normed space and $X_{1}$ be a subspace of $X$. Can one always extend a linear operator $l:X_{1}\rightarrow X$ to $L:X\rightarrow X$ such that $L$ has the same operator norm as $l$? Thanks!
Are all such groups solvable?
Are all groups of order $156$ solvable?
I have an epimorphism $f:B_4\longrightarrow S_4$, from the braid group on 4 strands onto the symmetric group on 4 elements. Is it possible the kernel is not isomorphic to $P_4$, the pure braid group on 4 strands?
Braid Group, B_4->> S_4 onto, do I know kernel is P_4, pure braid group?
Let $X$ be be a subspace of $\ell_1^4$. Can one always extend a linear operator $l:X\rightarrow \ell_1^4$ to $L:\ell_1^4\rightarrow \ell_1^4$ such that $L$ has the same operator norm as $l$? Thanks!
I want to construct a nowhere dense set of positive measure by using the Smith-Volterra-Cantor method but subtracting closed sets intead of open sets. Suppose we start with the open interval (0,1). Let $\alpha=1/2$. Now remove $U_1=[1/2-\alpha/4,1/2+\alpha/4]$. The measure $\mu(U_1)=\alpha/2$. For each of the tw...
"Does this give me a set of positive measure just as with subtracting closed intervals?" Yes, as follows from the basic property of [continuity of measure][1]: if $E_1 \supset E_2 \supset \ldots \supset E_n \supset \ldots$ is a nested sequence of subsets with $\mu(E_1) < \infty$, then if $E = \bigcap_{n=1}^{\infty} ...
"Does this give me a set of positive measure just as with subtracting closed intervals?" Yes, as follows from the basic property of [continuity of measure][1]: if $E_1 \supset E_2 \supset \ldots \supset E_n \supset \ldots$ is a nested sequence of subsets with $\mu(E_1) < \infty$, then if $E = \bigcap_{n=1}^{\infty} ...
How two check if two multiplications are equal to each other or greater or lesser without actually multiplying them? For example, compare (254)(847) and (383)(536) please correct the tags if wrong
How to Compare two multiplications without multiplying?
How to check if two multiplications are equal to each other or greater or lesser without actually multiplying them? For example, compare (254)(847) and (383)(536)
Thank you very much! My problem is: If R is a commutative ring with identity, and a, b are its elements that are divisible by each other, is it true that they must be associates? Here, a being divisible by b means there exists an r \in R, such that a=rb; and a and b being associates means there exists a inver...
If R is a commutative ring with identity, and a, b are its elements that are divisible by each other, is it true that they must be associates?
I'm trying to prove an identity involving the [digamma function](http://en.wikipedia.org/wiki/Digamma_function) $\psi(z)$, but I can't seem to figure out a way to do it. Can anyone help me out? The identity is $$\psi\left(\frac{m}{2} + iy\right) + \psi\left(\frac{m}{2} - iy\right) = \psi\left(\frac{n}{2} + iy\right)...
How can I prove this identity involving the digamma function?
I'm trying to prove an identity involving the [digamma function](http://en.wikipedia.org/wiki/Digamma_function) $\psi(z)$, but I can't seem to figure out a way to do it. Can anyone help me out? The identity is $$\psi\left(\frac{m}{2} + iy\right) + \psi\left(\frac{m}{2} - iy\right) = \psi\left(\frac{n}{2} + iy\right)...
See the following paper, When are Associates Unit Multiples? D.D. Anderson, M. Axtell, S.J. Forman, and Joe Stickles Rocky Mountain J. Math. Volume 34, Number 3 (2004), 811-828. This is [freely available][1]. It is mostly concerned with finding sufficent conditions on commutative ringsthat ensure that $Ra=R...
Let $a$ and $b$ be elements of the polynomial ring $\mathbb{Z}/n\mathbb{Z}[x]$. If $a$ and $b$ generate the same ideal, must they be associates? If $n$ is prime, then it is easy to see that the answer is "yes". But what if $n$ is not prime?
Is the functor from the category of projective varieties over a field $k$ to the category of pure motives over $k$, faithful? (Perhaps it is not full). Ditto: Is the functor from the category of affine varieties over a field $k$ to the category of mixed motives over $k$, faithful? (Perhaps it is not full). ...
Does a motive capture everything about an algebraic variety?
As Rasmus points out, once you state the result correctly, it's easy -assuming you've already proved some elementary properties of homotopy. Namely, let $f,g$ be two paths with same origin and end points: $f(0) = x_0 = g(0)$ and $f(1)= x_1 = g(1)$ and such that $g^{-1}*f \sim c_0$, where $c_0$ is the constant path a...
Let $\psi_{j,k}(t)= \left \{ \begin{array}{ll} 2^{j/2} &, 2^{-j}k < t < 2^{-j}(k+1/2) \\-2^{j/2} &, 2^{-j}(k+1/2) < t < 2^{-j}(k+1) \\ 0, \textrm{otherwise} \end{array} \right$. How to prove that it is orthogonal? In other words how to prove that $\langle \psi_{j_1,k_1}(t), \psi_{j_2, k_2}(t) \rangle = 0$, $j_1<j_2 an...

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